Indices
Basic Rules of Indices
- Any number raised to the power of 1 is the number itself, e.g: a^1 = a
- Any number raised to the power of 0 is 1, e.g: a^0 = 1
- If two terms with the same base are multiplied together, the powers are added, e.g: a^m * a^n = a^(m+n)
- If a term with a base of "a" raised to a power "m" is itself raised to a power "n", multiply the powers, e.g: (a^m)^n = a^(mn)
- If two terms with the same base are divided, subtract the exponent of the denominator from the exponent of the numerator, e.g: a^m / a^n = a^(m-n)
- If a term in the denominator has a negative exponent, it can be moved to the numerator and made positive, e.g: 1 / a^-n = a^n
- The n-th root of a number "a" can be denoted by a ^ (1/n)
Laws of Indices Involving Fractions
- a^-n = 1 / a^n, which means a reciprocal of a number can be written as a negative exponent.
- The n-th root of a number can be expressed as a power with a fractional exponent, e.g: n√a = a^(1/n)
- If the power of a term is a fraction, the denominator of the fraction is the root, and the numerator is the power. For example, a^(m/n) = ( n√a ) ^m
Simplification Using Indices
- In order to simplify expressions with indices, use the laws of indices to combine terms.
- Simplify the expression step by step, until no further simplification is possible.
- When dealing with algebraic expressions, it's important to remember that these rules apply only to terms with the same base.
Indices and Surds
- A Surd is an expression that includes a root (√). Surds are dealt with using the laws of indices.
- The method of 'rationalising the denominator' is used to remove surds from the denominator of a fraction.
- To simplify a surd, look for the largest square number which divides into the number under the surd, and use the rule √ab = √a * √b.
Exponential Equations
- Exponential equations are those where the variable is in the exponent.
- They can be solved using the principle that if a^m = a^n, then m = n.
- In cases where this cannot be applied directly, logarithms may be used to solve the equation.